Analysis of symmetries in models of multi-strain infections

Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: a bifurcation analysis

We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov-Takens and zero-Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks

We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with Z_N symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.

Chimera states in multi-strain epidemic models with temporary immunity

We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibit emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.

Mathematical analysis of a multiple strain, multi-locus-allele system for antigenically variable infectious diseases revisited

Many important pathogens such as HIV/AIDS, influenza, malaria, dengue and meningitis generally exist in phenotypically distinct serotypes that compete for hosts. Models used to study these diseases appear as meta-population systems. Herein, we revisit one of the multiple strain models that have been used to investigate the dynamics of infectious diseases with co-circulating serotypes or strains, and provide analytical results underlying the numerical investigations. In particular, we establish the necessary conditions for the local asymptotic stability of the steady states and for the existence of oscillatory behaviors via Hopf bifurcation. In addition, we show that the existence of discrete antigenic forms among pathogens can either fully or partially self-organize, where (i) strains exhibit no strain structures and coexist or (ii) antigenic variants sort into non-overlapping or minimally overlapping clusters that either undergo the principle of competitive exclusion exhibiting discrete strain structures, or co-exist cyclically.